Theorem recexprlemm 7623

Description: B is inhabited. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.

Assertion

Ref	Expression
recexprlemm	|- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )

Distinct variable groups:   r, q, x, y, A   B, q, r, x, y

Proof of Theorem recexprlemm

Step	Hyp	Ref	Expression
1		prop 7474	. . 3 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7477	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 2nd ` A ) )
3		recclnq 7391	. . . . . . 7 |- ( x e. Q. -> ( *Q ` x ) e. Q. )
4		nsmallnqq 7411	. . . . . . 7 |- ( ( *Q ` x ) e. Q. -> E. q e. Q. q <Q ( *Q ` x ) )
5	3, 4	syl 14	. . . . . 6 |- ( x e. Q. -> E. q e. Q. q <Q ( *Q ` x ) )
6	5	adantr 276	. . . . 5 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> E. q e. Q. q <Q ( *Q ` x ) )
7		recrecnq 7393	. . . . . . . . . . . 12 |- ( x e. Q. -> ( *Q ` ( *Q ` x ) ) = x )
8	7	eleq1d 2246	. . . . . . . . . . 11 |- ( x e. Q. -> ( ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) <-> x e. ( 2nd ` A ) ) )
9	8	anbi2d 464	. . . . . . . . . 10 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) <-> ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) ) )
10		breq2 4008	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( q <Q y <-> q <Q ( *Q ` x ) ) )
11		fveq2 5516	. . . . . . . . . . . . . 14 |- ( y = ( *Q ` x ) -> ( *Q ` y ) = ( *Q ` ( *Q ` x ) ) )
12	11	eleq1d 2246	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( ( *Q ` y ) e. ( 2nd ` A ) <-> ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) )
13	10, 12	anbi12d 473	. . . . . . . . . . . 12 |- ( y = ( *Q ` x ) -> ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) ) )
14	13	spcegv 2826	. . . . . . . . . . 11 |- ( ( *Q ` x ) e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
15	3, 14	syl 14	. . . . . . . . . 10 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
16	9, 15	sylbird 170	. . . . . . . . 9 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
17		recexpr.1	. . . . . . . . . 10 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
18	17	recexprlemell 7621	. . . . . . . . 9 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
19	16, 18	imbitrrdi 162	. . . . . . . 8 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) -> q e. ( 1st ` B ) ) )
20	19	expcomd 1441	. . . . . . 7 |- ( x e. Q. -> ( x e. ( 2nd ` A ) -> ( q <Q ( *Q ` x ) -> q e. ( 1st ` B ) ) ) )
21	20	imp 124	. . . . . 6 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> ( q <Q ( *Q ` x ) -> q e. ( 1st ` B ) ) )
22	21	reximdv 2578	. . . . 5 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> ( E. q e. Q. q <Q ( *Q ` x ) -> E. q e. Q. q e. ( 1st ` B ) ) )
23	6, 22	mpd 13	. . . 4 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> E. q e. Q. q e. ( 1st ` B ) )
24	23	rexlimiva 2589	. . 3 |- ( E. x e. Q. x e. ( 2nd ` A ) -> E. q e. Q. q e. ( 1st ` B ) )
25	1, 2, 24	3syl 17	. 2 |- ( A e. P. -> E. q e. Q. q e. ( 1st ` B ) )
26		prml 7476	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 1st ` A ) )
27		1nq 7365	. . . . . . . 8 |- 1Q e. Q.
28		addclnq 7374	. . . . . . . 8 |- ( ( ( *Q ` x ) e. Q. /\ 1Q e. Q. ) -> ( ( *Q ` x ) +Q 1Q ) e. Q. )
29	3, 27, 28	sylancl 413	. . . . . . 7 |- ( x e. Q. -> ( ( *Q ` x ) +Q 1Q ) e. Q. )
30		ltaddnq 7406	. . . . . . . 8 |- ( ( ( *Q ` x ) e. Q. /\ 1Q e. Q. ) -> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) )
31	3, 27, 30	sylancl 413	. . . . . . 7 |- ( x e. Q. -> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) )
32		breq2 4008	. . . . . . . 8 |- ( r = ( ( *Q ` x ) +Q 1Q ) -> ( ( *Q ` x ) <Q r <-> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) ) )
33	32	rspcev 2842	. . . . . . 7 |- ( ( ( ( *Q ` x ) +Q 1Q ) e. Q. /\ ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) ) -> E. r e. Q. ( *Q ` x ) <Q r )
34	29, 31, 33	syl2anc 411	. . . . . 6 |- ( x e. Q. -> E. r e. Q. ( *Q ` x ) <Q r )
35	34	adantr 276	. . . . 5 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> E. r e. Q. ( *Q ` x ) <Q r )
36	7	eleq1d 2246	. . . . . . . . . . 11 |- ( x e. Q. -> ( ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) <-> x e. ( 1st ` A ) ) )
37	36	anbi2d 464	. . . . . . . . . 10 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) <-> ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) ) )
38		breq1 4007	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( y <Q r <-> ( *Q ` x ) <Q r ) )
39	11	eleq1d 2246	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) )
40	38, 39	anbi12d 473	. . . . . . . . . . . 12 |- ( y = ( *Q ` x ) -> ( ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) ) )
41	40	spcegv 2826	. . . . . . . . . . 11 |- ( ( *Q ` x ) e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
42	3, 41	syl 14	. . . . . . . . . 10 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
43	37, 42	sylbird 170	. . . . . . . . 9 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
44	17	recexprlemelu 7622	. . . . . . . . 9 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
45	43, 44	imbitrrdi 162	. . . . . . . 8 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) -> r e. ( 2nd ` B ) ) )
46	45	expcomd 1441	. . . . . . 7 |- ( x e. Q. -> ( x e. ( 1st ` A ) -> ( ( *Q ` x ) <Q r -> r e. ( 2nd ` B ) ) ) )
47	46	imp 124	. . . . . 6 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> ( ( *Q ` x ) <Q r -> r e. ( 2nd ` B ) ) )
48	47	reximdv 2578	. . . . 5 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> ( E. r e. Q. ( *Q ` x ) <Q r -> E. r e. Q. r e. ( 2nd ` B ) ) )
49	35, 48	mpd 13	. . . 4 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> E. r e. Q. r e. ( 2nd ` B ) )
50	49	rexlimiva 2589	. . 3 |- ( E. x e. Q. x e. ( 1st ` A ) -> E. r e. Q. r e. ( 2nd ` B ) )
51	1, 26, 50	3syl 17	. 2 |- ( A e. P. -> E. r e. Q. r e. ( 2nd ` B ) )
52	25, 51	jca 306	1 |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   E.wrex 2456   <.cop 3596   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279   1Qc1q 7280   +Q cplq 7281   *Qcrq 7283   <Q cltq 7284   P.cnp 7290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-inp 7465

This theorem is referenced by:  recexprlempr  7631